On probability generating functions for waiting time distributions of compound patterns in a sequence of multistate trials

2002 ◽  
Vol 39 (01) ◽  
pp. 70-80 ◽  
Author(s):  
James C. Fu ◽  
Y. M. Chang
Keyword(s):  
Markov Chain ◽  
Waiting Time ◽  
Generating Functions ◽  
Probability Generating Function ◽  
Finite Markov Chain Imbedding ◽  
Waiting Time Distributions ◽  
Probability Generating Functions ◽  
Markov Chain Imbedding ◽  
Markov Chain Imbedding Method ◽  
Runs And Patterns

Probability generation functions of waiting time distributions of runs and patterns have been used successfully in various areas of statistics and applied probability. In this paper, we provide a simple way to obtain the probability generating functions for waiting time distributions of compound patterns by using the finite Markov chain imbedding method. We also study the characters of waiting time distributions for compound patterns. A computer algorithm based on Markov chain imbedding technique has been developed for automatically computing the distribution, probability generating function, and mean of waiting time for a compound pattern.

On probability generating functions for waiting time distributions of compound patterns in a sequence of multistate trials

2002 ◽  
Vol 39 (1) ◽  
pp. 70-80 ◽  
Author(s):  
James C. Fu ◽  
Y. M. Chang
Keyword(s):  
Markov Chain ◽  
Waiting Time ◽  
Generating Functions ◽  
Probability Generating Function ◽  
Finite Markov Chain Imbedding ◽  
Waiting Time Distributions ◽  
Probability Generating Functions ◽  
Markov Chain Imbedding ◽  
Markov Chain Imbedding Method ◽  
Runs And Patterns

Probability generation functions of waiting time distributions of runs and patterns have been used successfully in various areas of statistics and applied probability. In this paper, we provide a simple way to obtain the probability generating functions for waiting time distributions of compound patterns by using the finite Markov chain imbedding method. We also study the characters of waiting time distributions for compound patterns. A computer algorithm based on Markov chain imbedding technique has been developed for automatically computing the distribution, probability generating function, and mean of waiting time for a compound pattern.

Waiting times for patterns in a sequence of multistate trials

2001 ◽  
Vol 38 (2) ◽  
pp. 508-518 ◽  
Author(s):  
Demetrios L. Antzoulakos
Keyword(s):  
Markov Chain ◽  
Waiting Time ◽  
Waiting Times ◽  
Random Variable ◽  
Finite Markov Chain ◽  
Finite Markov Chain Imbedding ◽  
Markov Chain Imbedding ◽  
Markov Chain Imbedding Method ◽  
Runs And Patterns ◽  
Imbedding Method

Let Xn, n ≥ 1 be a sequence of trials taking values in a given set A, let ∊ be a pattern (simple or compound), and let Xr,∊ be a random variable denoting the waiting time for the rth occurrence of ∊. In the present article a finite Markov chain imbedding method is developed for the study of Xr,∊ in the case of the non-overlapping and overlapping way of counting runs and patterns. Several extensions and generalizations are also discussed.

Waiting times for patterns in a sequence of multistate trials

2001 ◽  
Vol 38 (02) ◽  
pp. 508-518 ◽  
Author(s):  
Demetrios L. Antzoulakos
Keyword(s):  
Markov Chain ◽  
Waiting Time ◽  
Waiting Times ◽  
Random Variable ◽  
Finite Markov Chain ◽  
Finite Markov Chain Imbedding ◽  
Markov Chain Imbedding ◽  
Markov Chain Imbedding Method ◽  
Runs And Patterns ◽  
Imbedding Method

Let X n , n ≥ 1 be a sequence of trials taking values in a given set A, let ∊ be a pattern (simple or compound), and let X r,∊ be a random variable denoting the waiting time for the rth occurrence of ∊. In the present article a finite Markov chain imbedding method is developed for the study of X r,∊ in the case of the non-overlapping and overlapping way of counting runs and patterns. Several extensions and generalizations are also discussed.

On ordered series and later waiting time distributions in a sequence of Markov dependent multistate trials

2003 ◽  
Vol 40 (3) ◽  
pp. 623-642 ◽  
Author(s):  
James C. Fu ◽  
Yung-Ming Chang
Keyword(s):  
Markov Chain ◽  
Waiting Time ◽  
Generating Functions ◽  
Finite Markov Chain ◽  
Finite Markov Chain Imbedding ◽  
Waiting Time Distributions ◽  
Markov Chain Imbedding ◽  
Exact Distributions ◽  
Markov Chain Imbedding Technique ◽  
Comprehensive Study

The sooner and later waiting time problems have been extensively studied and applied in various areas of statistics and applied probability. In this paper, we give a comprehensive study of ordered series and later waiting time distributions of a number of simple patterns with respect to nonoverlapping and overlapping counting schemes in a sequence of Markov dependent multistate trials. Exact distributions and probability generating functions are derived by using the finite Markov chain imbedding technique. Examples are given to illustrate our results.

On ordered series and later waiting time distributions in a sequence of Markov dependent multistate trials

2003 ◽  
Vol 40 (03) ◽  
pp. 623-642 ◽  
Author(s):  
James C. Fu ◽  
Yung-Ming Chang
Keyword(s):  
Markov Chain ◽  
Waiting Time ◽  
Generating Functions ◽  
Finite Markov Chain ◽  
Finite Markov Chain Imbedding ◽  
Waiting Time Distributions ◽  
Markov Chain Imbedding ◽  
Exact Distributions ◽  
Markov Chain Imbedding Technique ◽  
Comprehensive Study

The sooner and later waiting time problems have been extensively studied and applied in various areas of statistics and applied probability. In this paper, we give a comprehensive study of ordered series and later waiting time distributions of a number of simple patterns with respect to nonoverlapping and overlapping counting schemes in a sequence of Markov dependent multistate trials. Exact distributions and probability generating functions are derived by using the finite Markov chain imbedding technique. Examples are given to illustrate our results.

scholarly journals Distributions of Patterns of Pair of Successes Separated by Failure Runs of Length at Least and at Most Involving Markov Dependent Trials: GERT Approach

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Kanwar Sen ◽  
Pooja Mohan ◽  
Manju Lata Agarwal
Keyword(s):  
Waiting Time ◽  
Generating Functions ◽  
Waiting Times ◽  
Waiting Time Distributions ◽  
Probability Generating Functions ◽  
Mean And Variance ◽  
Markov Dependent Trials

We use the Graphical Evaluation and Review Technique (GERT) to obtain probability generating functions of the waiting time distributions of 1st, and th nonoverlapping and overlapping occurrences of the pattern , involving homogenous Markov dependent trials. GERT besides providing visual picture of the system helps to analyze the system in a less inductive manner. Mean and variance of the waiting times of the occurrence of the patterns have also been obtained. Some earlier results existing in literature have been shown to be particular cases of these results.

Polynomial bounds for probability generating functions

1975 ◽  
Vol 12 (3) ◽  
pp. 507-514 ◽  
Author(s):  
Henry Braun
Keyword(s):  
Lower Bound ◽  
Generating Function ◽  
Generating Functions ◽  
Branching Process ◽  
Probability Generating Function ◽  
Extinction Time ◽  
Probability Generating Functions ◽  
Extinction Probabilities ◽  
Polynomial Bounds

The problem of approximating an arbitrary probability generating function (p.g.f.) by a polynomial is considered. It is shown that if the coefficients rj are chosen so that LN(·) agrees with g(·) to k derivatives at s = 1 and to (N – k) derivatives at s = 0, then LN is in fact an upper or lower bound to g; the nature of the bound depends only on k and not on N. Application of the results to the problems of finding bounds for extinction probabilities, extinction time distributions and moments of branching process distributions are examined.

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